QM/MM models for solvated and embedded systems (ground state energies, properties and reactivity)

Workshop on Theoretical Chemistry: Modelling of Environment Effects in Quantum Chemistry

Mariapfarr, 20-23 February 2018Scientific Organizer: Benedetta Mennucci

Jeremy Harvey, KU Leuven, BelgiumLecture 1: Introduction to QM/MM

Multiscale Phenomena in Chemistry

JNH / Lecture 1 2

• Chemists need to think about the relation between microscopic properties and macroscale observables

• Usually this implies combining a model for a molecule- or atom-level property with a model for its impact at the macroscale

• E.g. water molecules form strong hydrogen bonds à boiling point of water much higher than that of H2S

• There are many intermediate length scales: supramolecular constructs (e.g. proteins/micelles), cells, etc

• Also there are many time scales: molecular periodic motions (electrons, vibrations) à millions of years

Multiscale Models of Different Types• The most accurate theoretical models in chemistry tend to be

applicable only for small molecules (and short timescales)• This is because electronic structure theory methods typically

scale steeply in terms of computational effort: N3 or worse• Size of phase space (number of conformers, timescale for

reaching equilibrium in simulations) grows very steeply with N• Modelling is of necessity always in one sense multi-scale:

quantum chemistry is used to predict molecular level properties, and a separate theory (e.g. statistical mechanics) is used to predict or rationalize macroscale observables

• In a fully-fledged multiscale model, a computer program is used to carry out integrated modelling of multiple scales

An Aside: Molecular Mechanics Force-FieldsIn molecular mechanics, the potential energy of the system for a given arrangement of nuclei is obtained as a ‘mechanical’ expression based on the energy required to distort bond lengths & angles, distort dihedrals, and allow non-bonded atoms to interact via. Coulomb’s Law + Lennard-Jones terms:

θi is an angle, φi a dihedral angle, and kij, ki, rij0, θi0, Ai and ni are parameters.

E.g. a dimer of water molecules: the sum includes 4 O–H stretching terms and 2 H–O–H bending terms

A Commonly Used Multiscale Model: QM/MM

5

The most fundamental description of molecular systems is based on nuclei and electrons (quantum mechanics, QM). In QM/MM, this level is used for a small “QM” region, with the bulk of the system described as atoms with bonds (molecular mechanics, MM)

MM is efficient à QM/MM applicable with thousandsof atoms, only treating the detailedelectronic structure of 10-200 atoms inthe QM region.

There are also QM/QM methods, with an accurate QM method for the core, anda faster one for the environment.

MMQM

Hybrid Methods: Hamiltonian

JNH / Lecture 1 6

The overall energy includesan energy term for both subsystemsand a coupling term:

BA

V = V A

high

+ V B

low

+ V A-B

High-level energy of A Low-level energy of B

This also yields (in principle) an electronic wave-function for the whole system:

= A�� A

high

⇥ B

low

↵

Hybrid Methods: Diversity

JNH / Lecture 1 7

Within the above framework, thereare very many variations on how totreat A and B, and the coupling betweenthem.

One frequently use distinction is between additive methods and subtractive ones. Another distinction is between QM/QM and QM/MM schemes. For QM/MM methods, there can be mechanical embedding or electronic embedding. Finally the treatment of boundary region bonds can be quite diverse.

BA

V = V A

high

+ V B

low

+ V A-B

Additive/Substractive MethodsIn additive methods, the energy of A is computed at one level of theory, thatof B at another, and the coupling betweenthem at a third level.Much diversity: nature of the coupling term (and to some extent the treatment of the environment B) can vary a lotSubtractive methods treat the whole system at the lower level of theory, and correct this at thehigh level for the active region:This can also be viewed as a high level calculation on the model, to which is added a corrective environment term:

BA

V = V AB

low

+ (V A

high

� V A

low

)

V = V A

high

+ (V AB

low

� V A

low

)

ONIOM (NB: original form)

JNH / Lecture 1 9

One of the most commonly used subtractive methods is ONIOM: “Our own N-layered Integrated molecular Orbital and molecular Mechanics” method.

ONIOM: A Multilayered Integrated MO + MM Method for Geometry Optimizations and Single Point Energy Predictions. A Test for Diels-Alder Reactions and Pt(P(t-Bu)3)2 + H2 Oxidative AdditionM. Svensson, S. Humbel, R. D. J. Froese, T. Matsubara, S. Sieber and K. MorokumaJ. Phys. Chem. 1996, 100, 19357-19363.

ONIOM is potentially n-layered, hence the acronym: like an onion, there can be many successive layers

ONIOM: Gradients

JNH / Lecture 1 10

The gradient (and higher derivatives) of the ONIOM energy can be straightforwardly computed from the gradients of the different methods

@V

@X=

@V A

high

@X+ (

@V AB

low

@X+

�@V A

low

@X)

ONIOM: Application

JNH / Lecture 1 11

J. Phys. Chem. 1996, 100, 19357-19363.

transition state has been predicted theoretically,11 while forasymmetric reactants, the transition states are also asymmetricwith the two forming bonds often differing distinctly in theirlengths.12,13 It has been shown that high levels of electroncorrelation are required to obtain activation energies quantita-tively in agreement with experimental values.13 Owing to strongelectronic and correlation effects, the Hartree-Fock methodbadly overestimates these activation energies, and MP2 resultsare often far too low. Our interest in using the ONIOM schemefor this type of reaction comes from the difficulty in obtaininggood agreement with experimental results at modest levels ofelectron correlation and the possible quantitative evaluation ofthe activation energy for larger systems where steric effects areimportant.In this section, we will apply the ONIOM3 method, as well

as IMOMM and IMOMO methods, to three Diels-Alderreactions: acrolein + isoprene (2-methyl-1,3-butadiene), ac-rolein + 2-tert-butyl-1,3-butadiene, and ethylene + 1,4-di-tert-butyl-1,3-butadiene. For the first two reactions, the results arecompared with experimental results and for the third withbenchmark calculations. Geometry optimizations will be per-formed at a variety of combinations of levels in ONIOM3,IMOMO, and IMOMM as well as in pure B3LYP and HF.Higher level single point energies will also be determined. Theintegrated methods should allow us to more easily separateelectronic and steric effects and to achieve high accuracy at afraction of the cost.A. Geometry Optimization of Acrolein + Isoprene Sys-

tem. For acrolein + isoprene and acrolein + 2-tert-butyl-1,3-butadiene reactions, we have divided our real system into threelayers as shown in Figure 2. The small model (SModel),ethylene + butadiene, with six non-H atoms is to describe theelectron reorganization during the reaction and is treated at thehighest level of approximation. The intermediate model (IMo-del), acrolein+ butadiene, with eight non-H atoms is to describethe electronic effects on the electron reorganization and is treatedat a medium level, and the full real system including thesubstituent (methyl or tert-butyl) at the butadiene 2-position isto describe mainly the steric effects and is treated at the lowestlevel.In these reactions, four products and corresponding transition

states are possible with the carbonyl group at the endo or exoposition with respect to the diene and with the alkyl substituenton the diene closer to or further from the carbonyl group.13,14

We examined the four transition states in the Diels-Alder [4+ 2] addition of acrolein to isoprene with the B3LYP method.It was determined that the structure with the lowest energy hasthe endo carbonyl and the isoprene methyl group away fromthe carbonyl group. In the following calculations, we consideredonly this regio- and stereospecific transition state.To test the reliability of the ONIOM scheme for the

optimization of geometries, we compared the B3LYP/6-31Ggeometry (which is considered as our benchmark geometry) tothe ONIOM (B3LYP/6-31G:HF/6-31G:MM3) geometries asshown in Figure 2. Additional IMOMM (B3LYP/6-31G:MM3:MM3) results are also presented here for comparison.The benchmark B3LYP/6-31G geometry is very asymmetric

with the two forming C-C bonds being 2.587 Å (adjacent tothe aldehyde group) and 2.049 Å (next to the methyl group). Inthe ethylene + butadiene case at the same level of theory, thesevalues are 2.265 Å for both sides. The reason for this strongasymmetry clearly is the electronic effects of the aldehyde group.The effects of the methyl group are not significant to thegeometry of this transition state. As a matter of fact, the fourpossible transition states for this reaction all exhibit nearly thesame geometry with these two distances being 2.56 ( 0.03 Åon the side of the aldehyde and 2.06 ( 0.01 Å on the oppositeside, regardless of the position of the methyl group. TheB3LYP/6-31G//B3LYP/6-31G barrier from trans-1,3-butadiene,19.8 kcal/mol, compares well with the experimental value of18.7( 0.8 kcal/mol,15 giving credibility to our benchmark result.The ONIOM3(B3LYP:HF:MM3) optimized structure in

Figure 2 looks quite reasonable as compared to the pure MOone. A single point B3LYP/6-31G energy at this ONIOMoptimized geometry is only 1.3 kcal/mol higher than that at thebenchmark-optimized structure. Since optimized reactants arealso higher by the same value, the B3LYP/6-31G activationenergy obtained at the ONIOM3 optimized geometry is exactlythe same (19.8 kcal/mol) as that at the benchmark-optimizedgeometry. The present results show that the ONIOM 3(B3LYP:HF:MM3) is capable of properly handling the steric effect ofthe “nonactive” part and the electronic effect of the “semiactive”carbonyl group.For comparison, the geometry optimization at the cheapest

IMOMM (B3LYP:MM3:MM3) method appears to almostcompletely miss the strong asymmetry of the transition state,leading to a nearly symmetric transition state with the formingC-C distances of 2.282 and 2.239 Å vs the benchmark valuesof 2.587 and 2.049 Å, respectively. The potential energy surfacenear the transition state is rather flat, and the energy of thisIMOMM optimized structure is 3.3 kcal/mol higher than thebenchmark, leading to a B3LYP/6-31G//IMOMM barrier of 21.8kcal/mol or 2 kcal/mol higher than the benchmark. ThisIMOMM scheme fails to account for the electronic effects ofthe carbonyl group, especially at the transition state.B. Single Point Energy Calculations for Acrolein +

Isoprene Diels-Alder Reaction. As mentioned above, theenergy barriers in Diels-Alder reactions are difficult to evaluatequantitatively for any system involving more than six non-Hatoms, since very high correlation levels, such as CCSD(T),are required, and this is very computer time intensive. However,using the IMOMO scheme,3,4 one can mimic very high levelsof calculation by treating only a small model very accuratelyand assuming the reliability of a modest correlation level, suchas MP2, in describing the real r model difference. In thissection, we will perform and compare the results of single pointenergy calculations with many different ONIOM integration

Figure 2. Small model, intermediate model, and real system for theDiels-Alder reactions between acrolein and 2-alkyl-1,3-butadiene. Theoptimized transition state bond distances (in Å) for the forming bondsare shown for the reaction of acrolein with isoprene and 2-tert-butyl-1,3-butadiene (to the right of /) at the nonintegrated (B3LYP:B3LYP:B3LYP) level in bold, those at the ONIOM3(B3LYP:HF:MM3) innormal, and those at the IMOMM(B3LYP:MM3:MM3) level in italic.

ONIOM J. Phys. Chem., Vol. 100, No. 50, 1996 19359

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schemes for the activation energy of the acrolein + isoprenereaction, all at the ONIOM3(B3LYP:HF:MM3) optimizedgeometries.As shown in Table 1, compared with the B3LYP/6-31G(d)

benchmark barrier height of 19.2 kcal/mol, the pure HF valueis more than two times too large. So is the pure MM3 value,if we dare to consider that MM3 can handle the early transitionstate assuming the maximum MM conjugation discussed above.A two-layered B3LYP:MM3:MM3 approach results in anactivation energy of 21.9 kcal/mol, which is a drastic improve-ment over the pure HF method, accounting for the importanceof electron correlation effects in describing the reacting regionof the system but still too large by 2.7 kcal/mol relative to thefull B3LYP benchmark. Four schemes, B3LYP:HF:MM3,B3LYP:HF:HF, B3LYP:B3LYP:MM3, and B3LYP:B3LYP:HF,all give barrier heights to within 1.4 kcal/mol of the benchmarkvalue. This indicates clearly that at least the HF level is requiredto describe the electronic effect of the carbonyl group and theMM3 force field method is enough to describe the effect of themethyl substituent. The differences in the activation energyamong these four schemes and the benchmark show that an errorof 0.8 kcal/mol is introduced by going from HF to MM3 forthe low level and an error of 0.5 kcal/mol by going from B3LYPto HF for the medium level. The error introduced when themedium level goes from HF to MM3 is much larger, 4 kcal/mol, and is not acceptable. Considering the computational effortrequired and the accuracy achieved, one can easily recognize,especially for much larger real systems than those studied herein this test, that the ONIOM3(B3LYP:HF:MM3) accomplishesa nearly quantitative agreement with the B3LYP benchmark,with a very small additional cost over that for the medium levelcalculation for the intermediate model.As mentioned in the Introduction, the pure MP2 calculation

gives a barrier too low by 6 kcal/mol, as also seen in Table 1.The integration of various levels of ab initioMOmethods allowsreliable evaluation of the activation barrier. The highest levelone may practically use, the two-layered CCSD(T):MP2:MP2scheme, gives the activation energy within the error limit ofthe experiment. The three-layered integrated MO method,CCSD(T):MP2:HF, and the ONIOM3 (CCSD(T):MP2:MM3)introduce a deviation of 0.3 and 0.6 kcal/mol, respectively, butare still within a chemical accuracy from the best CCSD(T):MP2:MP2 method. Again as expected, the IMOMM schemeCCSD(T):MM3:MM3 is a total failure because of the missing

electronic effect for the carbonyl group. As to the computationaleffort, if ONIOM3(CCSD(T):MP2:MM3) took 19 h of worksta-tion time, CCSD(T):CCSD(T):MP2 would have taken 3 daysand the pure CCSD(T) about 40 days even for this modest 12non-H atom system. For larger systems, the time differencewould be much larger.C. Acrolein + 2-tert-Butyl-1,3-butadiene System. Using

an MM method for the low level in the ONIOM3 scheme, onecan perform geometry optimization and energy calculation atvirtually no additional cost for any large “nonactive” third layer.To examine the effect of increased steric repulsion in and bythe third layer substituents, we have replaced the 2-methyl ofisoprene by 2-tert-butyl and tested the ONIOM schemes forthe Diels-Alder reaction between acrolein and 2-tert-butyl-1,3-butadiene. As shown in Figure 2, the transition state here isslightly less asymmetric than that for 2-methyl. The MM3 usedas the medium level is again totally inadequate.By use of ONIOM3 geometries, single point calculations were

carried out for the activation energy at various integration levels,as shown in Table 1. The performance of various schemes issimilar to that for the acrolein/isoprene system. One requiresCCSD(T) or B3LYP as the high level, MP2 or HF as theintermediate level, and MM3 as acceptable for the low level inthe present dividing scheme.The difference in the barrier heights between the 2-tert-butyl

and 2-methyl systems at the best CCSD(T):MP2:MP2 level is2.7 kcal/mol. In all other acceptable levels of calculation, onefinds that this difference is reproduced within (0.5 kcal/mol.Though there is no experimental value of activation energy forthe acrolein/2-tert-butyl-1,3-butadiene reaction, activation ener-gies for these two dienes reacting with a different olefin, maleicanhydride, are known to be 12.2 and 6.5 kcal/mol for the2-methyl and 2-tert-butyl, respectively. Our prediction for thereactions of acrolein is consistent with this experiment.16Why is the barrier for the 2-tert-butyl system lower than for

the 2-methyl system? It should be noted at first that the cis-isomer is more stable than the trans-isomer for the reactant,2-tert-butyl-1,3-butadiene, and is used as the reference for theactivation energy. For 2-methyl-1,3-butadiene the trans-isomeris more stable and is used as the reference, since the barrier fortrans f cis isomerization, 7.3 kcal/mol, is smaller than thereaction barrier. Since the difference can be described by theMM3 method, it must be steric in origin. If one compares theactivation barrier from the cis-isomer where the steric energyis expected to be smaller, the activation barrier is 16.2 kcal/mol (at CCSD(T):MP2:MP2 level) for 2-methyl-1,3-butadieneand 16.9 kcal/mol for 2-tert-butyl-1,3-butadiene. This is to saythat the steric energy at the transition state relative to the cis-reactant is only 0.7 kcal/mol larger for 2-tert-butyl-1,3-butadienethan for 2-methyl-1,3-butadiene and is not much different. Thus,the reason the barrier is lower for 2-tert-butyl-1,3-butadiene isthat the reaction for this system is initiated from the intrinsicallyless stable cis-isomer, since the trans-isomer of the reactant hasa large steric energy of 4.8 kcal/mol and is not the most stableisomer of the two anymore.D. Ethylene + s-Trans trans,trans-1,4-Di-tert-butyl-1,3-

butadiene System. As a final example of Diels-Alder systems,the reaction of ethylene with s-trans trans,trans-1,4-di-tert-butyl-1,3-butadiene was compared with benchmark predictions at theMP4(SDQ) level. Geometry optimizations were performed atthe B3LYP/6-31G level. We partitioned this system into a smallmodel of ethylene + butadiene, an intermediate model ofethylene+ 2,4-hexadiene (s-trans trans,trans-1,4-di-methyl-1,3-butadiene), and the real system. Table 2 gives the activationenergies for various ONIOM schemes with MP4(SDQ) always

TABLE 1: Activation Barrier (in kcal/mol) for theDiels-Alder Reaction of Acrolein + Isoprene (2-Methyl) and2-tert-Butyl-1,3-butadiene (2-tert-Butyl) Calculated withVarious ONIOM Schemesa,b

high:med:low 2-methyl 2-tert-butylB3LYP:B3LYP:B3LYP 19.2 16.4B3LYP:B3LYP:HF 19.1 17.1B3LYP:B3LYP:MM3 18.3 15.4B3LYP:HF:HF 18.6 16.6B3LYP:HF:MM3 17.8 15.0B3LYP:MM3:MM3 21.9 18.6HF:HF:HF 41.1 39.7MM3:MM3:MM3 42.1 39.9CCSD(T):MP2:MP2 19.6 16.9CCSD(T):MP2:HF 19.9 18.9CCSD(T):MP2:MM3 19.0 17.2CCSD(T):MM3:MM3 24.7 22.6MP2:MP2:MP2 12.7 10.1a Acrolein + trans-dienes were assumed as the reactants. For 2-tert-

butyl-1,3-butadiene, the s-cis-isomer is actually more stable than thes-trans-isomer by 1.4 kcal/mol at the CCSD(T):MP2 level. b Thegeometries optimized at the ONIOM3 (B3LYP:HF:MM3) level areused. The 6-31G(d) basis set is used in all MO calculations.

19360 J. Phys. Chem., Vol. 100, No. 50, 1996 Svensson et al.

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ONIOM: Application 2

JNH / Lecture 1 12

• ONIOM is typically aimed atorganometallic cases:

J. Phys. Chem. 1996, 100, 19357-19363.

MM3 result is 1.1 kcal/mol better than the two-layered B3LYP:MM3:MM3; the electronic effect of the alkyl group is consid-ered to contribute to this improvement.For the insertion product, the steric repulsion between P(t-

Bu)3 ligands is much larger than in the transition state. Theuse of the HF method as the low level clearly underestimatesthe steric repulsion by several kcal/mol. The MM3 method asthe low level overestimates the steric repulsion and gives theproduct about 3.5 kcal/mol too low.It is noted that the relative computational costs for the most

cost effective two-layered B3LPY:MM3:MM3 method (withoutconsideration of the alkyl electronic effect) and the three-layeredB3LYP:HF:MM3 method (with the alkyl electronic effectincluded) are 80 and 24 times, respectively, smaller than thatfor the pure B3LYP calculation. The last calculation itself isusually considered to be very cost effective but would take 1day on a workstation for the energy only; with the ONIOM orIMOMM scheme, one can perform nearly as reliable a calcula-tion within 1 h or less.As mentioned above, the correlation effect is very important

for the energetics for the oxidative addition. Therefore, tocalculate a reliable value for the activation barrier and the energyof formation, one has to adopt the highest level of electroncorrelation using such a method as CCST(T), which would beprohibitive for the present 83 atom H2 + Pt(P(t-Bu)3)2 reactioneven on the fastest computers. We have performed anONIOM3(CCSD(T)/TZP:MP2/DZP:MM3) calculation on aworkstation, which we expect to predict the energetics ofreaction within a few kcal/mol. The predicted activation barrieris 14.2 kcal/mol, which is lower than those for pure B3LYPand ONIOM3(B3LYP:HF:MM3) calculations by 4.1 and 3.3kcal/mol, respectively. The major difference comes from thedifference between CCSD(T) and B3LYP results for the smallmodel system. The predicted endoergicity of reaction is 4.1kcal/mol, which is 6.4 and 2.0 kcal/mol smaller than the pureB3LYP and ONIOM3(B3LYP:HF:MM3) results. Since CCSD-(T) correlation energy with finite basis sets is usually anunderestimate, the barrier and the endoergicity could be a few

kcal/mol smaller. This single point ONIOM3(CCSD(T):MP2:MM3) calculation costs only 40% of the pure B3LYP calcula-tion.

V. Conclusions

ONIOM is a multilayered approach integrating various levelsof MO and MM methods in optimizing transition state andequilibrium geometries and in performing single point energycalculations. Its three-layer version ONIOM3 divides a systeminto an active, a semiactive, and a nonactive part and considersa small model, an intermediate model, and the real system. Thesmall model would be typically treated with a very high levelof correlation such as CCSD(T), the intermediate model with amodest accuracy MO level such as MP2 and HF, and the realsystem with low accuracy and inexpensive method such as MMforce fields. The motivation behind ONIOM is the costeffectiveness, with the high-level method the most importantpart and handled very accurately, and at the same time theelectronic effects of the semiactive and the steric and electro-static effects of the nonactive part would taken into account ata very small cost.In the example of the Diels-Alder reaction of acrolein +

substituted butadiene it has been shown that the two-layeredIMOMM method using ethylene + butadiene fails because theelectronic effect of the carbonyl group is not taken into account.On the other hand, one would not like to spend a large amountof computer time to include the steric effects of bulky substit-uents. Inserting a third layer at the modest MP2 or HF levelinto the IMOMM approach gives a better description of theelectronic effects and can reproduce the results of a very high-level calculation with an error of a few kcal/mol.With this method, it would become possible to make

predictions of the activation energy or the bond dissociationenergy of a very large system within a chemical accuracy of afew kcal/mol. Using the ONIOM3(CCSD(T):MP2:MM3)method, we have predicted the activation barrier and the energyof reaction for the 83 atom H2 + Pt(P(t-Bu)3)2. The costs ofthe ONIOM3(B3LPY:HF:MM3) and the ONIOM3(CCSD(T):MP2:MM3) calculations for this system are about 4% and 40%,respectively, of the pure B3LYP calculation.

Acknowledgment. The authors are grateful to Dr. JamalMusaev and Qiang Cui for their numerous help and discussions.The use of the Emerson Center computing facilities is acknowl-edged. The present research is in part supported by grants(CHE-9409020 and CHE-9627775) from the National ScienceFoundation. R.D.J.F. acknowledges a postdoctoral fellowshipfrom the Natural Sciences and Engineering Research Councilof Canada. M.S. acknowledges a postdoctoral fellowship fromthe Swedish Natural Science Research Council.

References and Notes(1) Field, M. J.; Bash, P. A.; Karplus, M. J. Comput. Chem. 1990, 11,

700.(2) Maseras, F.; Morokuma, K. J. Comput. Chem. 1995, 16, 1170.(3) Humbel S.; Sieber S.; Morokuma, K. J. Chem. Phys. 1996, 105,

1959.(4) Svensson M.; Humbel S.; Morokuma, K. J. Chem. Phys. 1996, 105,

3654.(5) Matsubara, T.; Maseras, F.; Koga, N.; Morokuma, K. J. Phys. Chem.

1996, 100, 2573.(6) The code we implemented combines the IMOMM code to the

former IMOMO code. For the IMOMM code, see ref 2 and for the IMOMOsee ref 3.

(7) Frisch, M. J.; Trucks, G. W.; Head-Gordon, M.; Gill, P. M. W.;Wong, M. W.; Foresman, J. B.; Johnson, B. G.; Schelgel, H. B.; Robb, M.A.; Replogle, E. S.; Gomperts, R.; Andres, J. L.; Raghavachari, K.; Binkley,

TABLE 3: Activation Barriers Ea, Energies of Reaction Er(in kcal/mol), and Their Errors from the Pure B3LYPBenchmark for the Oxidative Addition of H2 to Pt(P(t-Bu)3)2Calculated with Various ONIOM Schemesa

Ea ¢Ea Er ¢Ercomputertimeb

B3LYP:B3LYP:B3LYPc 18.3 0.0 10.5 0.0 1207HF:HF:HFd 24.6 6.3 18.7 8.2 438B3LYP:B3LYP:HF 19.1 0.8 14.9 4.4 586B3LYP:B3LYP:MM3 16.8 -1.5 7.0 -3.5 148B3LYP:HF:HF 19.8 1.5 14.0 3.4 453B3LYP:HF:MM3 17.5 -0.8 6.1 -4.4 51B3LYP:MM3:MM3 16.4 -1.9 8.0 -2.5 15CCSD(T) e:MP2f:MM3 14.2 4.1 500

a The geometries optimized at the IMOMM(MP2:MM3) level takenfrom ref 5 are used. The energy of reaction is calculated for the cis-product. b The relative computer time required. c Basis set: (5s5p3d1f/3s3p2d1f) with RECP of Hay and Wadt for Pt (Hay, P. J.; Wadt, W.R. J. Chem. Phys. 1985, 82, 299), (4s1p/2s1p) for active H2, and adouble-˙ set for P, C, and H on P(t-Bu)3 (Dunning, T. H.; Hay, P. J.In Modern Theoretical Chemistry; Schaefer, H. F., III, Ed.; Plenum:New York, 1976; p 1). d Basis set: the same as in footnote c abovebut without polarization functions. e Basis set: (5s5p3d1f/4s4p3d1f)with RECP of Hay and Wadt for Pt, 6-311G(2p) for active H2, and thesame as in footnote c augmented with 1d(0.55) on P and 1p(0.75) onH for PH3. f Basis set: (5s5p3d1f/3s3p2d1f) with RECP of Hay andWadt for Pt, 6-31G(p) for active H2, and the same as in footnote c forP (augmented with 1d(0.55)) and C and 6-31G for H for P(t-Bu)3.

19362 J. Phys. Chem., Vol. 100, No. 50, 1996 Svensson et al.

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ONIOM: Link Atoms

JNH / Lecture 1 13

• The energy for the sub-system A is not well defined when there is a covalent bond crossing the boundary. Instead, in such cases, A is augmented by an appropriate number of link atoms (typically H atoms).

J. Phys. Chem. 1996, 100, 19357-19363.

Here an H atom is addedto convert CH2OH into CH3OH

High-level region

ONIOM: Link Atoms and Gradients

JNH / Lecture 1 14

How do gradients work?

L C

@V

@X=

@V A

high

@X+ (

@V AB

low

@X+

�@V A

low

@X)

Feliu Maseras and Keiji Morokuma, J. Comp. Chem. 1995, 16, 1170-1179

Vhigh implicitly depends on the coordinates XC of atom ‘replaced’ by link atom L, as XL itself depends on XC (e.g. L is positioned along the same bond axis)so that:

=)@V A

high

@XC=

@V Ahigh

@XL⇥ @XL

@XC

XL �XQ = ↵⇥ {XC �XQ}

ONIOM: Mechanical Embedding

JNH / Lecture 1 15

When the low level in ONIOM is a forcefield method, the only impact the environment has on the subsystem is to distort it according to the coupling terms implicit in the correction term:

Especially where the forcefield containsno point charges (ball and spring modeltypical for hydrocarbons), this is apurely “mechanical” effect.

V = V A

high

+ (V AB

low

� V A

low

)

QM/MM and Electronic Embedding

JNH / Lecture 1 16

For some systems, electrostatic interactions between the QM and MM regions can be quite strong – and the MM environment may be expected to polarize the QM region somewhat. For these systems, it is desirable to allow the QM wavefunction to relax with respect to the MM point charges. This leads to typical QM/MM electronic embedding methods.

V = V AQM + V B

MM + V A-BQM-MM

rq

qq

q

q

q

q

q

q

q

q

q

Electronic Embedding in Detail

JNH / Lecture 1 17

In practice the energy expression is rather:

rq

qq

q

q

q

q

q

q

q

q

q

V = V ABQM’ + V AB

MM’

V ABQM’ is the QM energy of A in the presence of the point charges

(n electrons, N nuclei,NMM MM point charges)

V ABMM’ is more straightforward:

it contains the MM energyof MM atoms + MM-likeinteractions between MM and QM atoms

Helec =�1

2

nX

i=1

r2i +

nX

i=1

NX

A=1

�ZA

riA+

nX

i=1

nX

j>i

1

rij+

nX

i=1

NMMX

K=1

�qKriK

Electronic Polarization

JNH / Lecture 1 18

Point charges indeed lead to similar polarization compared to all-QM calculations:

Analysis of polarization in QM/MM modelling of biologically relevant hydrogen bonds, Senthilkumar, Mujika, Ranaghan, Manby, Mulholland and Harvey, J. Roy. Soc. Interface, 2008, 5, S207 – S216

nX

i=1

NMMX

K=1

�qKriK

7!nX

i=1

NMMX

K=1

(�qK ⇥ f(riK))

Charge Leakage

JNH / Lecture 1 19

Regions of high electron density in the QM region could in principle ‘leak out’ onto positive point charges due to lack of Pauli repulsion by these charges. Not a big problem for modest-sized Gaussian basis sets, but significant with plane waves.

Solution: replace 1/r term with ashielded interaction.

See e.g. A regularized and renormalized electrostatic coupling Hamiltonian for hybrid quantum-mechanical–molecular-mechanical calculations, P. K. Biswas and V. Gogonea, J. Chem. Phys. 2005, 123, 164114.

Covalent bonds and the QM-MM boundary

JNH / Lecture 1 20

In QM/MM calculations, covalent bonds between the QM and MM boundary can be treated in several different ways:• Link atoms, as in mechanical embedding

Note that the electrostatic treatment of MM atoms near the MM boundary can be varied, and this can have a big effect on results

• Pseudo-atoms: instead of an H atom to replace the bonded MM atom, use some kind of 1-electron atom with a pseudopotential designed to yield the correct bond length and electronic structure. (See discussion in Xiao and Zhang, J. Chem. Phys. 2007, 127, 124102)

• Frozen orbitals: constrain one hybrid atomic orbital of the QM atom to have exactly the same shape it has in a reference compound, with two electrons – behaves a bit like a lone pair (see e.g. Murphy, Philipp and Friesner, J. Comp. Chem. 2000, 21, 1442.)

QM/MM Gradients

JNH / Lecture 1 21

There are four types of gradient elements in QM/MM with electronic embedding:

@VQM

@XQM

@VQM

@XMM

@VMM

@XQM

@VMM

@XMM

Like standard QM gradientexcept for modified w.f.

Not routinely computed by QMcodes; but for SCF methods,just equal to –qK × EQM

Standard MM-like terms, forinteractions (e.g. vdW) with theQM atoms

Standard MM

Exploring QM/MM Potential Energy Surfaces

JNH / Lecture 1 22

• Large systems place high demands on the Hamiltonian• QM/MM as well as (near-)linear scaling methods can now

routinely cope with (very) large systems• Nowadays, a larger challenge of big systems is the complexity of

their potential energy surfaces• Can be explored using geometry optimization and reaction path

methods• Or by simulation with QM/MM• Free energies from Umbrella Sampling/Metadynamics and other

related approaches• Combining MM and QM/MM or QMlow/MM and QMhigh/MM

QM/MM Geometry Optimization: Microiterations

JNH / Lecture 1 23

Time-consuming for large systems (N atoms). Even efficient algo-rithms tend to take ~N steps until acceptable convergence.Even if each energy and gradient evaluation for the whole QM/MM system is cheap (roughly the cost of the QM energy + gradient for the QM region), this makes optimization expensive.Hence microiterations: carry out many cycles of optimization of MM atom positions for each calculation of QM energy/gradient. Possible because QM and MM degrees of freedom are almost uncoupled:@VMM

@XMM

Does not depend onthe QM system

Quite small; depends only weakly onthe QM wavefunction – can be appro-ximated by electrostatics.

@VQM

@XMM

See e.g. A Mixed Quantum Mechanics/Molecular Mechanics (QM/MM) Method for Large-Scale Modeling of Chemistry in Protein Environments, Murphy, Philipp and Friesner, J. Comp. Chem. 2000, 21, 1442.

Energy Minimization Reaches Local Minima

JNH / Lecture 1 24

• Starting from a given structure,one reaches a local minimum (typically the ‘closest’ one)

• A system with N atoms typicallyhas a potential energy surface with ~exp(αN) local minima – a large number!!

• There are also large numbers of saddle points• One typically cannot find the global minimum, or the lowest

energy saddlepoint – unlike for systems with ~10s of atoms• This is not a difficulty limited to QM/MM – it is rather a

property of large systems (MM, semiempirical, large QM)

CA CBCC

CD

Characterizing Minima

JNH / Lecture 1 25

• For molecular systems, optimization is typically followed by calculating frequencies.

• These require the Hessian, i.e. the terms:

@V 2QM

@Xi@Xj

@V 2MM

@Xi@Xj

@V

2MM

@xi@Xj

@V

2MM

@xi@xj

@V

2QM

@xi@xj

@V

2QM

@xi@Xj

Full and partial Hessian

JNH / Lecture 1 26

• The full Hessian may be unavailable, or too large to store or diagonalize (e.g. 10,000 atoms => 30,000 x 30,000)

• Also, typically in QM/MM, some atoms may be frozen during optimization => typically one does not have a minimum

• Gradient norm not arbitrarily close to zero, so ‘frequencies’ for rotational modes may not be that close to zero.

• It can be useful to diagonalize a block sub-Hessian (see Efficient Calculation of QM/MM Frequencies with the Mobile Block Hessian, Ghysels et al., JCTC 2011, 7, 496-514.)

• Provided that eigenvectors for vibrational modes of interest (e.g. the reaction coordinate) are well within the block of atoms for which the Hessian is generated, then should be OK.

Reaction path study• In molecular systems, a typical app-

roach is to locate relevant minimafor reactants, intermediates,products, then TSs separating them

• This can be done also with QM/MM: algorithms for TS searching can be used more or less as such for QM/MM systems (using e.g. a partial block Hessian to guide the search)

• Two difficulties: many coordinates,and hard to guarantee that the located minimum and TS areconnected

• The variance in energies of local minima / TSstypically exceeds the target relative energy.

TSATSB TSC TSD

CA CBCC

CD

Reaction paths: Adiabatic mapping

JNH / Lecture 1 28

• In order to obtain meaningful rel-ative energies along e.g. a reactionpath, one can use techniques suchas adiabatic mapping

• Key coordinate (here ‘q2’) chosen• Minima found for set of values of

this coordinate q2,i (enforced e.g. byadding harmonic term k(q2-q2,i)2.

• This method works well when thereaction path remains highly parallel to q2 along the whole reaction – but in practice reaction path curvature is a major issue

Nudged Elastic BandsImproved tangent estimate in the nudged elas-tic band method for finding minimum energypaths and saddle points, Henkelmann and Jónsson, J. Chem. Phys. 2000, 113, 9978-9985

For each image, one follows a modifiedgradient, the sum of a harmonic springforce in the direction tangent to reac-tion path, and the true gradient orthogonal to the tangent. This tangent direction is given by:

minima had dropped and the spacing between them in-creased enough to satisfy the stability condition, Eq. !6".Then the images at the minima were slowly pulled into placeby the spring force. In this way, chains with up to #80images were able to !slowly" converge. For more images, theminimization would not converge and some images werealways left in a jumble at the minima.

In order to further test the stability condition, the restor-ing force perpendicular to the path was doubled by settingAy!2. The stability condition predicts that a band with twicethe number of images, 24, will remain stable. Calculationsshowed that the band becomes unstable at 21 images. Thegood agreement between the simple prediction of Eq. !6" andthese simulations suggest that this is the correct origin of thekinks. The modified NEB method with the new tangent, pre-sented in the next section, converges well for both small andlarge numbers of images.

IV. THE NEW IMPLEMENTATION OF NEB

By using a different definition of the local tangent atimage i, the kinks can be eliminated. Instead of using boththe adjacent images, i"1 and i#1, only the image withhigher energy and the image i are used in the estimate. Thenew tangent, which replaces Eq. !2", is

!i!! !i" if Vi"1$Vi$Vi#1

!i# if Vi"1%Vi%Vi#1

, !8"

where

!i"!Ri"1#Ri , and !i

#!Ri#Ri#1 , !9"

and Vi is the energy of image i, V(Ri). If both of the adja-cent images are either lower in energy, or both are higher inenergy than image i, the tangent is taken to be a weightedaverage of the vectors to the two neighboring images. Theweight is determined from the energy. The weighted averageonly plays a role at extrema along the MEP and it serves tosmoothly switch between the two possible tangents !i

" and!i

# . Otherwise, there is an abrupt change in the tangent asone image becomes higher in energy than another and thiscan lead to convergence problems. If image i is at a mini-mum Vi"1$Vi%Vi#1 or at a maximum Vi"1%Vi$Vi#1 ,the tangent estimate becomes

!i!! !i"$Vi

max"!i#$Vi

min if Vi"1$Vi#1

!i"$Vi

min"!i#$Vi

max if Vi"1%Vi#1, !10"

where

$Vimax!max! "Vi"1#Vi", "Vi#1#Vi""

and !11"

$Vimin!min! "Vi"1#Vi", "Vi#1#Vi"".

Finally, the tangent vector needs to be normalized. With thismodified tangent, the elastic band is well behaved and con-verges rigorously to the MEP if sufficient number of imagesare included in the band.

Another small change from the original implementationof the NEB is to evaluate the spring force as

Fis" #!k! "Ri"1#Ri"#"Ri#Ri#1""!i !12"

instead of Eq. !5". This ensures equal spacing of the images!when the same spring constant, k, is used for the springs"even in regions of high curvature where the angle betweenRi#Ri#1 and Ri"1#Ri is large.

When this modified NEB method is applied to the sys-tem of Fig. 1 the band is well behaved as shown in Fig. 4.The energy and force of the NEB images is shown in Fig. 5along with the exact MEP obtained by moving along thegradient from the saddle point. The thin line though thepoints is an interpolation which involves both the energy andthe force along the elastic band. The details of the interpola-tion procedure is discussed in the Appendix.

The motivation for this modified tangent came from astable method for finding the MEP from a given saddle point.A good way to do this is to displace the system by someincrement from the saddle point and then minimize the en-ergy while keeping the distance between the system and thesaddle point configuration fixed. This gives one more pointalong the MEP, say M 1 . Then, the system is displaced againby some increment and minimized keeping the distance tothe point M 1 fixed, etc. The important fact is that the MEPcan be found by following force lines down the potentialfrom the saddle point, but never up from a minimum. If aforce line is followed up from a minimum, it will most likelynot go close to the saddle point. This motivated the choicefor the local tangent to be determined by the higher energyneighboring image in the NEB method.

A. Exchange diffusion process in Si crystalA particularly severe problem with kinks had been no-

ticed in calculations of self diffusion in a Si crystal. Forexample, one possible mechanism is the exchange of twoatoms on adjacent lattice sites.20 Both calculations usingDFT and empirical potentials could not converge the forcebecause of kinks. In calculations using 16 images and theTersoff potential,19 the force fluctuations remained at a level

FIG. 4. With the modified tangent, Eqs. !8"–!12", the nudged elastic bandmethod does not develop kinks and converges smoothly to the minimumenergy path !solid line".

9981J. Chem. Phys., Vol. 113, No. 22, 8 December 2000 Finding minimum energy paths and saddle points

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X0, X1, X2, . . . , Xn�1, Xn

⌧i =Xi �Xi�1

|Xi �Xi�1|+

Xi+1 �Xi

|Xi+1 �Xi|

Lecture 1 Conclusions

JNH / Lecture 1 30

• The QM/MM method provides an attractive and efficient method to compute energies and wavefunctions for large systems

• Many tests suggest that QM/MM is reasonably accurate for describing many physical effects at the QM:MM boundary

• The gradient (and higher derivatives) of the energy are well defined, so geometry optimization is possible

• Not because we use a QM/MM hamiltonian, but simply due to the large size of the typical systems studied, geometry optimization has some new difficulties compared to smaller (‘QM’) systems

• Global vs. Local minima• Connectivity of reaction paths: need adiabatic mapping or NEB